Contents:
Scalars
Scalars are defined solely by magnitude. Examples of scalars are physical quantities such as length, time, and mass.
Vectors
Vectors are defined by both magnitude and direction. Examples of physical vector components are velocity, momentum and force. Vectors are typically defined by bold text or an arrow above the symbol denoting the given vector. Vector components are defined by a 3 by 1 matrix, denoting the x, y, and z components of the vector.
\bm{\overrightarrow{a}}=\begin{bmatrix}a_x\\a_y\\a_z\end{bmatrix}
The magnitude of a vector is defined as:
a=\left|\bm{\overrightarrow{a}}\right|=\sqrt{a^2_x+a^2_y+a^2_z}
In two dimensions we will only concern ourselves with the x and y components, understanding that the z component is zero.
\bm{\overrightarrow{a}}=\begin{bmatrix}a_x\\a_y\end{bmatrix}
In two dimensions vector components can also be determined using the angle between the vector and the x-axis, as well as the vector magnitude.
a=\left|\bm{\overrightarrow{a}}\right|=\sqrt{a^2_x+a^2_y}
a_x=a\cos\theta
a_y=a\sin\theta
Vector Operations
Vectors can be added by simply adding the vector components individually:
\bm{\overrightarrow{a}}+\bm{\overrightarrow{b}}=\begin{bmatrix}a_x\\a_y\\a_z\end{bmatrix}+\begin{bmatrix}b_x\\b_y\\b_z\end{bmatrix}=\begin{bmatrix}a_x+b_x\\a_y+b_y\\a_z+b_z\end{bmatrix}
There are two forms of vector multiplication, dot product and cross product. The dot (or scalar) product can be thought of as the projection of one vector onto another. The cosine of the angle between both vectors can be used to calculate the dot product:
\bm{\overrightarrow{a}}\cdot \bm{\overrightarrow{b}}=\left|\bm{\overrightarrow{a}}\right|\left|\bm{\overrightarrow{b}}\right|\cos\theta
\bm{\overrightarrow{a}}\cdot \bm{\overrightarrow{b}}=a_xb_x+a_yb_y+a_zb_z
The cross product yields a vector perpendicular to the two input vectors. The direction of the output vector is dependent on the order in which the input vectors are multiplied (reversing the order results in a vector of the same magnitude in the opposite direction). The magnitude of the cross product is related to the sine of the angle between both input vectors:
\left|\bm{\overrightarrow{a}}\times \bm{\overrightarrow{b}}\right|=\left|\bm{\overrightarrow{a}}\right|\left|\bm{\overrightarrow{b}}\right|\sin\theta
\bm{\overrightarrow{a}}\times\bm{\overrightarrow{b}}=\begin{vmatrix}\bm{\hat{i}}&\bm{\hat{j}}&\bm{\hat{k}}\\a_x&a_y&a_z\\b_x&b_y&b_z\end{vmatrix}=\begin{vmatrix}a_y&a_z\\b_y&b_z\end{vmatrix}\bm{\hat{i}}-\begin{vmatrix}a_x&a_z\\b_x&b_z\end{vmatrix}\bm{\hat{j}}+\begin{vmatrix}a_x&a_y\\b_x&b_y\end{vmatrix}\bm{\hat{k}}
\bm{\overrightarrow{a}}\times\bm{\overrightarrow{b}}=\begin{bmatrix}a_yb_z-a_zb_y\\a_zb_x-a_xb_z\\a_xb_y-a_yb_x\end{bmatrix}